Understanding "mathematical thinking" and "the essence of the problem"
Math, as shown here, is a domain where if you understand the essence of the problem, you can always find a much simpler answer through an evolved method. This is exactly what I most want to emphasize to you through the first problem of this book.
In math, understanding the essence of a problem is far more important than calculation — that is, if you grasp the essence, you can extract a general form that can be applied whenever, and as a result, problem-solving becomes much easier and simpler.
Someone may ask back, "What use would math be in my life now, of all things?" But I think math is exactly the discipline best suited to pursuing the essence of a problem — that is, the principle of wisdom. And if you can understand and see through the essence, you can come up with ideas, or explain concepts to others more easily. Is there anything else that helps us as much in navigating life? (p.22)
Tsunenobu Okabe, translated by Kim Jeong-hwan — "Re-reading Math at Forty
— Meeting the Principle of Wisdom at Life's Halfway Point" (Yein (PlutoBook))
Try out this math problem for old times' sake.
"Problem: Suppose utility poles are installed along the Earth's equator. When wires are strung at a height of 10 meters, how much longer is the total length of the wires than the circumference of the equator? Assume the wires are taut with no sag. Use the Earth's radius as about 6,370 km and pi as 3.14."
How would you answer? The solution that first comes to mind is this.
Using the formula for the circumference of a circle, first calculate the length of the equator. Equator length = Earth's diameter (radius X 2) X pi (3.14), so 6,370,000 X 2 X 3.14 = 40,003,600 meters.
Next, calculate the total length of the wire on the utility poles. Since the wire is placed at 10 meters in height, just add 10 meters to the Earth's radius.
So (6,370,000 + 10) X 2 X 3.14 = 6,370,010 X 2 X 3.14 = 40,003,662.8 meters.
Therefore, 40,003,662.8 - 40,003,600 = 62.8 meters is the answer.
But the author, Dr. Tsunenobu Okabe, says that writing out numbers like this one by one inevitably makes calculations complicated, and advises bringing in "mathematical thinking" in such cases. Just assume, whatever the Earth's radius is, to call it A.
Then the Earth's diameter becomes 2 X A, and the wire's diameter becomes 2 X (A + 10).
Then the answer can be obtained this way.
3.14 X 2 X (A + 10) - 3.14 X 2 X A
= 3.14 X 2 X A + 3.14 X 2 X 10 - 3.14 X 2 X A
= 3.14 X 2 X 10
= 62.8 meters.
Swapping the Earth's radius from a number to A made the calculation process much simpler. To find how much longer the wire's circumference is than the Earth's circumference, there was no need to work the radius in as a specific number. The author's advice — "understanding the essence of the problem is much more important than calculation" — rings true.
It brings to mind the anecdote about Gauss (1777-1855), often called one of the three greatest mathematicians in history, who at about seven years old was given the task of summing the numbers from 1 to 100 and reportedly finished in just a few seconds. Gauss did not solve the problem by adding faster than others; he looked at the characteristics of numbers and found a fast, simple way to compute — and that is why he found the answer so quickly.
Even long after leaving school, occasionally opening a math book to build "mathematical thinking" can help you grasp "the essence of the problem" in office or business settings.
▶ Yeh Byung-il's Economic Notes — Twitter: @yehbyungil / Facebook: www.facebook.com/yehbyungil
